arXiv 1306.48102013-06math.APmath.DG

一类ultraparabolic equations的矩阵微分Harnack估计

A matrix differential Harnack estimate for a class of ultraparabolic equations

Hong Huang

备注：: 13 pages, the condition of the main theorem is slightly weakened, and more details are added

摘要：

设$u$是超抛物方程\begin{equation*} \partial_t u=\sum_{i=1}^n \partial_{x_i}^2 u+\sum_{i=1}^k x_i\partial_{x_{n+i}}u \hspace{8mm} \mbox{on} \hspace{4mm} \mathbb{R}^{n+k}\times (0,T), \end{equation*}的正解，其中$1\leq k\leq n$和$0<T \leq +\infty$ .假设$u$及其衍生物（w.r.t.直到二阶的空间变量）在$(0,T)$的任何紧致子区间上有界。然后，将$\log u$和$\log f$的Hessian矩阵之差$H(\log u)- H(\log f)$（均为w.r.t.空间变量）是非负定的，其中$f$是上述方程的基本解，极点在原点$(0,0)$。$n=k=1$案例中的估算是由Hamilton做出的。作为推论，我们得到$Δl+\frac{n+3k}{2t}+\frac{6k}{t^3}\geq 0$，其中$l=\log u$和$Δ=\sum_{i=1}^{n+k} \partial_{x_i}^2 $。

英文摘要：

Let $u$ be a positive solution of the ultraparabolic equation \begin{equation*} \partial_t u=\sum_{i=1}^n \partial_{x_i}^2 u+\sum_{i=1}^k x_i\partial_{x_{n+i}}u \hspace{8mm} \mbox{on} \hspace{4mm} \mathbb{R}^{n+k}\times (0,T), \end{equation*} where $1\leq k\leq n$ and $0<T \leq +\infty$. Assume that $u$ and its derivatives (w.r.t. the space variables) up to the second order are bounded on any compact subinterval of $(0,T)$. Then the difference $H(\log u)- H(\log f)$ of the Hessian matrices of $\log u$ and of $\log f$ (both w.r.t. the space variables) is non-negatively definite, where $f$ is the fundamental solution of the above equation with pole at the origin $(0,0)$. The estimate in the case $n=k=1$ is due to Hamilton. As a corollary we get that $Δl+\frac{n+3k}{2t}+\frac{6k}{t^3}\geq 0$, where $l=\log u$, and $Δ=\sum_{i=1}^{n+k} \partial_{x_i}^2 $.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文